A heat exchanger used to heat a glycol solution with a hot

oil is known to exhibit FOPTD behavior, $G_1(s)=T^{\text{'}}\frac{s}{Q^{\text{'}}}(s),$

where $T^{\text{'}}$ is the outlet temperature deviation and $Q^{\text{'}}$ is the

hot oil flow rate deviation. A thermocouple is placed $3m$

downstream from the outlet of the heat exchanger. The aver$-$

age velocity of the glycol in the outlet pipe is $0\mathrm{.}5\frac{m}{s}.$ The

thermocouple also is known to exhibit first$-$order behavior;

however, its time constant is expected to be considerably

smaller than the heat exchanger time constant.

$($a$)$ Data from a unit step test in $Q^{\text{'}}$ on the complete system are

shown in Fig. E$7\mathrm{.}7.$ Using a method of your choice, calculate

the time constants of this process from the step response.

$($b$)$ From your empirical model, find transfer functions for the

heat exchanger, pipe, and thermocouple. Think of the model

as the product of three transfer functions: process, pipe flow,

and sensor$.$ What assumptions do you have to make to obtain

these individual transfer functions from the overall transfer

function?